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In 2008, Borovik and Cherlin posed the problem of showing that the degree ofgeneric transitivity of an infinite permutation group of finite Morley rank$(X,G)$ is at most $n+2$ where $n$ is the Morley rank of $X$. Moreover, theyconjectured that the bound is only achieved (assuming transitivity) by$\operatorname{PGL}_{n+1}(\mathbb{F})$ acting naturally on projective$n$-space. We solve the problem under the two additional hypotheses that (1)$(X,G)$ is $2$-transitive, and (2) $(X-\{x\},G_x)$ has a definable quotientequivalent to$(\mathbb{P}^{n-1}(\mathbb{F}),\operatorname{PGL}_{n}(\mathbb{F}))$. The latterhypothesis drives the construction of the underlying projective geometry and isat the heart of an inductive approach to the main problem.

Towards the recognition of $\operatorname{PGL}_n$ via a high degree of generic transitivity